Enhancement of Luminous Intensity Emission from Incoherent LED Light Sources within the Detection Angle of 10° Using Metalenses

In this work, we present metalenses (MLs) designed to enhance the luminous intensity of incoherent light-emitting diodes (LEDs) within the detection angles of 0° and 10°. The detection angle of 0° refers to the center of the LED. Because the light emitted from LEDs is incoherent and expressed as a surface light source, they are numerically described as a set of point sources and calculated using incoherent summation. The titanium dioxide (TiO2) and amorphous silicon (a-Si) nanohole meta-atoms are designed; however, the full 2π phase coverage is not reached. Nevertheless, because the phase modulation at the edge of the ML is important, an ML is successfully designed. The typical phase profile of the ML enhances the luminous intensity at the center, and the phase profile is modified to increase the luminous intensity in the target detection angle region. Far field simulations are conducted to calculate the luminous intensity after 25 m of propagation. We demonstrate an enhancement of the luminous intensity at the center by 8551% and 2115% using TiO2 and a-Si MLs, respectively. Meanwhile, the TiO2 and a-Si MLs with the modified phase profiles enhance the luminous intensity within the detection angle of 10° by 263% and 30%, respectively.


Light Source Design for Simulation
The LEDs are described as numerous point sources with Lambertian intensity profiles that are incoherently summed. Thus, the designed LED source is an incoherent, unpolarized, surface light source. A 35 µm square LED source is used in this study. A micro-LED chip can be considered as a point source [82]. Figure 1 shows the simulated results after 20 µm of propagation with different numbers of points sources. The intensity profiles converge as the number of points sources become 3 × 3. In addition, the optimized and required chip size is at least 7 µm for typical display panels, including smartphones, laptops, and televisions [83][84][85], so 5 × 5 and 7 × 7 micro-chips are not considered in this study. We consider the LED source composed of 3 × 3 micro-LED chips, and the light source is described by 3 × 3 points sources. Because the light emitted from the LED is partially spatial-coherent, the phase of light originating from each point source is random and therefore interference effects are ignored. The propagation results shown in Figure 1 are obtained by adding only the amplitudes of the propagated light.
The Lambertian intensity profile is expressed as [86] I(θ) = I 0 cos θ (1) where I is the intensity, θ is the propagation angle with respect to the optical axis, and I 0 is the maximum intensity at θ = 0, respectively. To describe the Lambertian point source, an interface between two different media is used ( Figure S1). Because the transmittance at the interface is dependent on θ, the intensity of the transmitted light also varies with θ. When the distance between the point source and the interface is 40 nm and the refractive indices of the media are 1 and 2, the intensity profile of the transmitted light is similar to a Lambertian intensity profile, therefore allowing us to describe the single point source. The LED source is then modeled as 3 × 3 point sources placed 40 nm away from the interface. The schematics of the simulation is shown in Figure S2. Figure S3 shows the difference when a single point source and 3 × 3 points sources are used. However, individual simulations are done for each point source as the LED source is partially spatial-coherent, and therefore interference must be neglected [87]. The final calculations result from the incoherent summation of each individual simulation. The total number of simulations can be reduced if the system is symmetrically designed, allowing some results to be obtained by rotating other results. In addition, the partial temporal coherence of the LED source provides a broad spectral bandwidth of emitted light [88]. The simulation results with both narrowband and broadband sources provide similar results ( Figure S4); therefore, the effect of the partial temporal coherence is negligible and ignored.  In this work, the points sources are placed at the center of the 3 × 3 lattice ( Figure S2b). The effect of the relative locations of the sources is negligible in a 3 µm × 3 µm LED source, as the 3 × 3 array of dipole sources provides a similar light emission to a single point source of the same size [89].

Phase Profile of Metalens
The typical phase profile ϕ t of an ML is expressed as [50] where r, λ, and f are the radial coordinate, the wavelength, and the focal length, respectively. Spherical waves from the LED source are transmitted through the ML and collimated. However, the collimated light is concentrated at the propagation angle of 0 • , so the phase profile is modified to enhance the luminous intensity within the detection angle of 10 • . The modified phase profile ϕ m is expressed as where L is the length of the ML, m is the number of point sources that make up of the LED source, and α is the detection angle. L is 35 µm, the same with the length of the LED source, m is 3 because 3 × 3 points sources are used, and α is 10 • in this study. Because the lengths of the ML and the LED are identical, an ML array can be used to cover a large-area LED composed of equally sized LED pixels. The phase profile from Equation (3) is obtained by adding the phase profiles of the typical ML (Equation (2)) and a concave lens ( Figure S5). The second term of Equation (3) originates from the concave lens allowing the transmitted light to spread out to the target detection angle region. Because the LED source is a surface light source and is described as a set of the points sources, the designed ML is split into the subsections one-to-one correspondence with the point sources and the phase profile in each subsection is defined by Equation (2) or (3). The phase profiles at f = 10 µm and λ = 560 nm are shown in Figure 2. The phase gradient of ϕ m becomes smaller at the edge of the ML than that of ϕ t . Thus, when ϕ m is used, the diverging light from the source is steered less and spreads out to the target detection angle region.

Nanohole Meta-Atom
Since LEDs emit unpolarized light, polarization-insensitive meta-atoms are designed. In addition, nanohole meta-atoms are used instead of nanofins due to good adhesion between the nanohole structure and the substrate when the high aspect ratio (above

Nanohole Meta-Atom
Since LEDs emit unpolarized light, polarization-insensitive meta-atoms are designed. In addition, nanohole meta-atoms are used instead of nanofins due to good adhesion between the nanohole structure and the substrate when the high aspect ratio (above 1:5) nanostructures are fabricated ( Figure S6). Because the dense material is continuously connected, the nanohole structure provides a high production yield. However, the nanofin structure can be relatively easily destroyed during the etching or cleaning process. When the optical axis of the system is parallel to the z-direction, the transmitted electric field through the nanohole is expressed as [50,[90][91][92]] where t l and t s are complex transmission coefficients when the incident light is polarized along the long and short axis of the nanohole, respectively, and β is the rotation angle of the nanohole with respect to the x-axis. The first and the second terms of Equation (4) are known as the co-and cross-polarization terms, respectively, because the Jones vector of each term shows that the polarization of the transmitted light is identical to the incident light at the first term and is opposite the incident light at the second term. The polarization dependence is caused by exp(±i2β) from the cross-polarization term. Therefore, the metaatoms are designed to have the same t l and t s to remove the cross-polarization term, which can be achieved using symmetrical meta-atoms such as circle and square holes ( Figure S7). Titanium dioxide (TiO 2 ) and amorphous silicon (a-Si) nanohole meta-atoms are usually used at visible frequencies due to their high refractive indices and low loss [93]. Gallium nitride (GaN) has also been used at visible frequencies; however, a sapphire substrate is required and the sapphire substrate is less transparent than fused silica [49]. Thus, TiO 2 and a-Si are used in this study. The transmission properties of TiO 2 nanohole metaatoms with a height of 400 nm are investigated using rigorous coupled-wave analysis (RCWA) simulations ( Figure 3) [94]. The designed meta-atoms work under the spherical wave incidence even though they are designed by RCWA under the plane wave incidence ( Figure S8). The phase shown in Figure 3 does not satisfy the complete 2π coverage. However, they can be used to construct an ML even though the meta-atom library does not fully cover the phase from −π to π [54]. Meanwhile, the transmission properties of a-Si nanohole meta-atoms with height of 600 nm are also calculated using RCWA simulations ( Figure 4). The fabrication process for a-Si meta atoms is relatively easier and is less restrictive than that of TiO 2 meta-atoms, so a-Si meta-atoms can be designed with higher aspect ratios. Nevertheless, the phase depicted in Figure 4 does not cover the entire 2π range. The averaged transmittance of a-Si meta-atoms is lower than that of TiO 2 meta-atoms. meta-atoms with height of 600 nm are also calculated using RCWA simulations ( Figure 4). The fabrication process for a-Si meta atoms is relatively easier and is less restrictive than that of TiO2 meta-atoms, so a-Si meta-atoms can be designed with higher aspect ratios. Nevertheless, the phase depicted in Figure 4 does not cover the entire 2π range. The averaged transmittance of a-Si meta-atoms is lower than that of TiO2 meta-atoms.   Figure 5 shows the phases of the designed TiO2 MLs. The phase profile at the center of the lens is somewhat different from the desired phases. However, the ϴ of light at the center is already below 10° ( Figure 6) and is smaller than 10° at r < 1.76 μm. Thus, the transmitted light can reach the target detection angle region regardless of the zero phase gradient at the center. The MLs should steer the incident light at the edge to enhance the intensity within the target detection angle of 10°. Therefore, beam control at the edge is enough to construct the MLs, and the phase modulation at the edge is important for these designs. meta-atoms with height of 600 nm are also calculated using RCWA simulations (Figure 4). The fabrication process for a-Si meta atoms is relatively easier and is less restrictive than that of TiO2 meta-atoms, so a-Si meta-atoms can be designed with higher aspect ratios. Nevertheless, the phase depicted in Figure 4 does not cover the entire 2π range. The averaged transmittance of a-Si meta-atoms is lower than that of TiO2 meta-atoms.   Figure 5 shows the phases of the designed TiO2 MLs. The phase profile at the center of the lens is somewhat different from the desired phases. However, the ϴ of light at the center is already below 10° ( Figure 6) and is smaller than 10° at r < 1.76 μm. Thus, the transmitted light can reach the target detection angle region regardless of the zero phase gradient at the center. The MLs should steer the incident light at the edge to enhance the intensity within the target detection angle of 10°. Therefore, beam control at the edge is enough to construct the MLs, and the phase modulation at the edge is important for these designs.  Figure 5 shows the phases of the designed TiO 2 MLs. The phase profile at the center of the lens is somewhat different from the desired phases. However, the θ of light at the center is already below 10 • (Figure 6) and is smaller than 10 • at r < 1.76 µm. Thus, the transmitted light can reach the target detection angle region regardless of the zero phase gradient at the center. The MLs should steer the incident light at the edge to enhance the intensity within the target detection angle of 10 • . Therefore, beam control at the edge is enough to construct the MLs, and the phase modulation at the edge is important for these designs. Nanomaterials 2022, 12, x FOR PEER REVIEW 7 of 14

Far Field Propagation Simulation
The diverging spherical wave from the LED source passes through the ML and is able to propagate to the far field as the ML collimates the transmitted light. The emitted light from the LED source is at a distance f from the ML, then propagates for 25 m. The simulated propagation results are shown in Figure 7. The luminous intensities are calculated. Each ML enhances the luminous intensity at the center and the luminous intensity profile shows the peak near the angle of 0°. The peaks of the MLs with φt at the center are higher but narrower than the MLs with φm regardless of the materials. Because the crosssectional intensity profiles shown in Figure 7 are radially symmetric, the intensity at large angles are more dominant to enhance the total luminous intensity. Table 1 shows the luminous intensities of the MLs with respect to the angle from 0° (center) to 10° (target detection angle). The MLs with φm have larger luminous intensities, except at the angle of 0°, than that of the MLs with φt for both materials. In addition, full widths at half maximum

Far Field Propagation Simulation
The diverging spherical wave from the LED source passes through the ML and is able to propagate to the far field as the ML collimates the transmitted light. The emitted light from the LED source is at a distance f from the ML, then propagates for 25 m. The simulated propagation results are shown in Figure 7. The luminous intensities are calculated. Each ML enhances the luminous intensity at the center and the luminous intensity profile shows the peak near the angle of 0°. The peaks of the MLs with φt at the center are higher but narrower than the MLs with φm regardless of the materials. Because the crosssectional intensity profiles shown in Figure 7 are radially symmetric, the intensity at large angles are more dominant to enhance the total luminous intensity. Table 1 shows the luminous intensities of the MLs with respect to the angle from 0° (center) to 10° (target detection angle). The MLs with φm have larger luminous intensities, except at the angle of 0°, than that of the MLs with φt for both materials. In addition, full widths at half maximum

Far Field Propagation Simulation
The diverging spherical wave from the LED source passes through the ML and is able to propagate to the far field as the ML collimates the transmitted light. The emitted light from the LED source is at a distance f from the ML, then propagates for 25 m. The simulated propagation results are shown in Figure 7. The luminous intensities are calculated. Each ML enhances the luminous intensity at the center and the luminous intensity profile shows the peak near the angle of 0 • . The peaks of the MLs with ϕ t at the center are higher but narrower than the MLs with ϕ m regardless of the materials. Because the cross-sectional intensity profiles shown in Figure 7 are radially symmetric, the intensity at large angles are more dominant to enhance the total luminous intensity. Table 1    The MLs enhance the peak intensity by steering the direction of the transmitted light but do not increase the light extraction efficiency of the LED. In addition, some portion of light is reflected or absorbed by the MLs. Therefore, the enhancement of the luminous intensity at the center implies that the intensity at the other areas decreases. The efficiencies of the MLs are shown in Figure 8 with respect to α. The efficiency is defined as the output luminous intensity within α ( Figure S9) divided by the luminous intensity of the  The MLs enhance the peak intensity by steering the direction of the transmitted light but do not increase the light extraction efficiency of the LED. In addition, some portion of light is reflected or absorbed by the MLs. Therefore, the enhancement of the luminous intensity at the center implies that the intensity at the other areas decreases. The efficiencies Nanomaterials 2022, 12, 153 9 of 14 of the MLs are shown in Figure 8 with respect to α. The efficiency is defined as the output luminous intensity within α ( Figure S9) divided by the luminous intensity of the LED source. The TiO 2 ML with ϕ m has higher efficiencies within α = 10 • , 20 • , and 30 • than those of the bare LED, but the ML with ϕ t has lower efficiency within α = 30 • . The luminous intensity within α = 10 • is enhanced by 234% and 263% compared with the LED source when the TiO 2 MLs with ϕ t and ϕ m are used, respectively (Table S1). Because the ML with ϕ m has the wide intensity peak profile, the ML has a higher luminous intensity within α = 10 • than that of the ML with ϕ t despite the lower maximum luminous intensity at the center. Meanwhile, the enhancement of the luminous intensity within α = 20 • and 30 • is not significant, because the enhancement within α = 10 • is a consequence of steering the transmitted light from the other area to the target detection angle region. The a-Si ML with ϕ m provides the higher efficiencies within α = 10 • , 20 • , and 30 • than those of the a-Si ML with ϕ t due to the wider intensity profile. However, the a-Si MLs enhance the efficiencies slightly within α = 10 • and have lower efficiencies within α = 20 • and 30 • due to the low transmittance of a-Si meta-atoms. Nanomaterials 2022, 12, x FOR PEER REVIEW 9 of 14 LED source. The TiO2 ML with φm has higher efficiencies within α = 10°, 20°, and 30° than those of the bare LED, but the ML with φt has lower efficiency within α = 30°. The luminous intensity within α = 10° is enhanced by 234% and 263% compared with the LED source when the TiO2 MLs with φt and φm are used, respectively (Table S1). Because the ML with φm has the wide intensity peak profile, the ML has a higher luminous intensity within α = 10° than that of the ML with φt despite the lower maximum luminous intensity at the center. Meanwhile, the enhancement of the luminous intensity within α = 20° and 30° is not significant, because the enhancement within α = 10° is a consequence of steering the transmitted light from the other area to the target detection angle region. The a-Si ML with φm provides the higher efficiencies within α = 10°, 20°, and 30° than those of the a-Si ML with φt due to the wider intensity profile. However, the a-Si MLs enhance the efficiencies slightly within α = 10° and have lower efficiencies within α = 20° and 30° due to the low transmittance of a-Si meta-atoms.

Conclusions
The MLs are designed to enhance the luminous intensity of incoherent LED sources after 25 m of propagation within α = 0° and 10°. The LED source is composed of 3 × 3 micro-chips and is designed as the set of 3 × 3 Lambertian point sources. The propagation results are obtained by incoherently adding the simulation result with each single point source. Polarization-insensitive nanohole meta-atoms are designed. Although the phase profiles of the designed MLs do not match with the required phases at the center, the emitted light from the LED source at the center already propagates in the target detection angle region, so the phase modulation at the edge of the MLs is important in this study. When the TiO2 and a-Si MLs with φt are used, the luminous intensity at the center is enhanced by 8551% and 2115%, respectively. Meanwhile, the TiO2 and a-Si MLs with φm enhance the luminous intensity within α = 10° by 263% and 30%, respectively. Because the TiO2 meta-atoms have higher transmittance than that of a-Si, the TiO2 ML shows the higher enhancement. Meanwhile, the efficiency enhancement within α = 20° or 30° is not significant because the ML cannot increase the light extraction efficiency of the LED source and the enhancement within α = 0° or 10° is a consequence of steering the transmitted light. Therefore, the emitted light from the LED source can be delivered over a long distance using the ML, proving that it can be employed for various practical applications using LEDs, including the automotive headlights, display panels and so on.

Conclusions
The MLs are designed to enhance the luminous intensity of incoherent LED sources after 25 m of propagation within α = 0 • and 10 • . The LED source is composed of 3 × 3 micro-chips and is designed as the set of 3 × 3 Lambertian point sources. The propagation results are obtained by incoherently adding the simulation result with each single point source. Polarization-insensitive nanohole meta-atoms are designed. Although the phase profiles of the designed MLs do not match with the required phases at the center, the emitted light from the LED source at the center already propagates in the target detection angle region, so the phase modulation at the edge of the MLs is important in this study. When the TiO 2 and a-Si MLs with ϕ t are used, the luminous intensity at the center is enhanced by 8551% and 2115%, respectively. Meanwhile, the TiO 2 and a-Si MLs with ϕ m enhance the luminous intensity within α = 10 • by 263% and 30%, respectively. Because the TiO 2 meta-atoms have higher transmittance than that of a-Si, the TiO 2 ML shows the higher enhancement. Meanwhile, the efficiency enhancement within α = 20 • or 30 • is not significant because the ML cannot increase the light extraction efficiency of the LED source and the enhancement within α = 0 • or 10 • is a consequence of steering the transmitted light. Therefore, the emitted light from the LED source can be delivered over a long distance using the ML, proving that it can be employed for various practical applications using LEDs, including the automotive headlights, display panels and so on.

Methods
Commercially available Lumerical FDTD and VirtualLab Fusion were used for the far field simulations. The electric field transmitted through the ML was calculated using Lumerical FDTD. Then the electric field data were imported to VirtualLab Fusion and used as the source for the far field propagation. Finally, the simulated far field propagation results from each individual point source were incoherently summed up.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/10 .3390/nano12010153/s1, Figure S1: Description of a point source with Lambertian intensity profile: (a) Schematics; (b) The Lambertian (black line) and simulated (red line) intensity profiles; Figure  S2: Schematics of the simulation in this work: (a) The simulation from the LED source to the monitor is carried out using Lumerical FDTD. The far field simulation from the monitor to the detector is conducted using VirtualLab Fusion; (b) Distribution of the 3 × 3 points sources; Figure  S3: Comparisons of the simulated results calculated from a single point source and 3 × 3 points sources; Figure S4: Comparisons of the simulated results with a narrowband source at λ = 560 nm and broadband source at 500 ≤ λ ≤ 620 nm; Figure S5: Description of the effect of the ML with ϕ m explained by geometrical optics; Figure S6: Good adhesion of nanohole meta-atoms: (a) a-Si nanostructure composed of nanofin meta-atoms; (b) a-Si nanostructure composed of nanohole meta-atoms. Scale bar: 1 µm; Figure S7. Comparison of t l (solid line) and t s (circle dot) of TiO 2 meta-atoms: (a) Circle nanohole meta-atoms; (b) Square nanohole meta-atoms; Figure S8: Simulation result of the collimation using the TiO 2 ML with ϕ t . The focal length is 3 µm. The phase is plotted instead of the intensity, because the intensity at the point source is extremely larger than that at the other area; Figure S9: Measurement of the intensity within a detection angle of α. α = 0 • or 10 • and l = 25 m; Table S1: Comparison of enhancement of intensity from LED sources of the previously reported results.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.